Method of determining high-speed VLSI reduced-order interconnect by non-symmetric lanczos algorithm

ABSTRACT

Two-sided projection-based model reductions has become a necessity for efficient interconnect modeling and simulations in VLSI design. In order to choose the order of the reduced system that can really reflect the essential dynamics of the original interconnect, the element of reduced model of the transfer function can be considered as a stopping criteria to terminate the non-symmetric Lanczos iteration process. Furthermore, it can be found that the approximate transfer function can also be expressed as the original interconnect model with some additive perturbations. The perturbation matrix only involves at most rank-2 modification at the previous step of the non-symmetric algorithm. The information of stopping criteria will provide a guideline for the order selection scheme used in the Lanczos model-order reduction algorithm.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates generally to a reduced-order circuitmodel, and more particularly to a rapid and accurate reduced-orderinterconnect circuit model which can be used for signal analysis ofhigh-speed and very-large IC interconnect.

2. Description of Related Art

With rapid development of semiconductor techniques, the parasitic effecthas no longer been ignored during design of high-speed and very-large ICinterconnect. This technology was proposed in 2002 by M. Celik, L. T.and A. Odabasioglu “IC Interconnect Analysis,” Kluwer AcademicPublisher.

Given the fact of more complex circuit, the corresponding order ofmathematical model will be increased in order to simulate accurately thecharacteristics of interconnect circuits. Therefore, an efficient modelreduction method has become a necessary know-how for interconnectmodeling and simulation. The well-proven technologies, such as U.S. Pat.Nos. 6,789,237, 6,687,658, 6,460,165, 6,135,649, 601,170, 6,023,573, areproposed in 2000 by R. W. Freund, “Krylov-Subspace Methods forReduced-Order Modeling in Circuit Simulation,” Journal of Computationaland Applied Mathematics, Vol. 123, pp. 395-421; in 2002 by J. M. Wang,C. C. Chu, Q. Yu and E. S. Khu, “On Projection Based Algorithms forModel Order Reduction of Interconnects,” IEEE Trans. on Circuits andSystems-I: Fundamental Theory and Applications, Vol. 49, No. 1, pp.1563-1585.

In recent years, the common methods for circuit model reduction include:

Asymptotic Waveform Evaluation (AWE)(L. T. Pillage and R. A. Rohrer,“Asymptotic waveform evaluation for timing analysis,” IEEE Trans. onComputer-Aided Design of Integrated Circuits and Systems, Vol. 9, No. 4,pp. 352-366, 1990);

PVL (Pade via Lanczos)(P. Feldmann and R. W. Freund, “Efficient linearcircuit analysis by Pad'e approximation via the Lanczos process,” IEEETrans. on Computer-Aided Design of Integrated Circuits and Systems, Vol.14 pp. 639-649, 1995);

SyMPVL (Symmetric Matrix Pade via Lanczos)(P. Feldmann and R. W. Freund,“The SyMPVL algorithm and its applications to interconnect simulation,”Proc. 1997 Int. Conf. on Simulation of Semiconductor Processes andDevices, pp. 113-116, 1997);

Arnoldi Algorithm (e.g. U.S. Pat. No. 6,810,506);

PRIMA (Passive Reduced-order Interconnect Macromodeling Algorithm)(A.Odabasioglu, M. Celik and L. T. Pileggi, “PRIMA: passive reduced-orderinterconnect macromodeling algorithm,” IEEE Trans. on Computer-AidedDesign of Integrated Circuits and Systems, Vol. 17 pp. 645-653, 1998).

All of the aforementioned model reduction techniques employ KrylovSubspace Projection Method, which utilizes projection operator to obtainthe state variables of reduced circuit system after projecting the statevariable of original circuit system. The projection operator isestablished by Krylov Algorithm iteration process, of which the order ofreduced circuit is the number of iteration. For model reductionalgorithm of applied projection method, another important job is todetermine the order of reduced circuit, since it is required to find outan appropriate order such that the reduced circuit can reflectaccurately important dynamic behavior of original circuit.

SUMMARY OF THE INVENTION

The present invention provides an improved non-symmetric LanczosAlgorithm. Based on error estimation of linear circuit is reduced modeland original model, it should thus be possible for improvement ofsubmicron IC interconnect model.

The present invention will present a detailed description of therelationship between original circuit system and reduced circuit system,of which the reduced circuit is to obtain project-based matrix and thenreduced-order circuit model by employing non-symmetric LanzcosAlgorithm. Based on δ_(q+1) and δ_(q+1) calculated by the algorithm, itis required to set a termination iteration condition in order to obtaina balance point between complexity of computation and accuracy ofreduced model.

In addition, the present invention will prove that, after transferfunction of original circuit is added with some additive perturbations,the moment of transfer function fully matches that of reduced model byemploying non-symmetric Lanczos Algorithm in various orders. Sincewell-proven technology has demonstrated that q-th moments of reducedsystem are equivalent to those of original system, so q-th order momentsof original system plus perturbed system are equivalent to those oforiginal system. Of which, perturbation matrix is related to componentgenerated by non-symmetric Lanczos Algorithm, with the cyclomatic numberup to 2, so no additional computational resources are required. Thealgorithm of the present invention will provide an efficient guidelineof selecting reduced-order circuit by Krylov Subspace Model ReductionAlgorithm.

In a certain embodiment, the present invention has simplified circuitmodel by employing non-symmetric Lanczos Algorithm, which includes thefollowing steps: (1) input a mesh circuit; (2) input an expand frequencypoint; (3) set up a state space matrix of circuit; (4) reduce submicronIC interconnect model by employing an improved non-symmetric LanczosAlgorithm.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows the pseudo code of the traditional non-symmetric LanczosAlgorithm.

FIG. 2 shows the flow process diagram of reduced circuit by employingnon-Symmetric Lanczos Algorithm.

FIG. 3 shows the pseudo code of improved non-Symmetric Lanczos Algorithm

FIG. 4 shows the simplified embodiment of the present invention.

FIG. 5 shows the curve diagram of termination conditions of the presentinvention.

FIG. 6 shows the frequency response diagram of simplified embodiment.

FIG. 7 shows the error analysis of reduced-order models of simplifiedembodiment.

FIG. 8 shows the analysis pattern of order of simplified embodiment andmoment value of system.

DESCRIPTION OF THE MAIN COMPONENTS

-   -   (102) Original system    -   (104) Reduced order q=1    -   (106) Bi-orthogonal v_(q) and w_(q) of non-symmetric Lanczos        Algorithm    -   (108) Solution of δ_(q+1), δ_(q+1) $\begin{matrix}        {\lambda_{q} = {{\frac{\delta_{q + 1}}{{AV}_{q}}} \leq {ɛ\quad{and}\quad{\frac{\beta_{q + 1}}{A^{\prime}W_{q}}}} \leq ɛ}} & (110)        \end{matrix}$    -   (112) q++    -   (114) Calculate the model of reduced system

DETAILED DESCRIPTION OF THE INVENTION

The conventional methods, such as Modified Nodal Analysis (MNA),Kirchhoff's Voltage Law (KVL) and Kirchhoff's Current Law (KCL), areused for analyzing the characteristics of very-large IC interconnects.The circuits can be expressed as the following state space matrixes:$\begin{matrix}{{{M\frac{\mathbb{d}{x(t)}}{\mathbb{d}t}} = {{- {{Nx}(t)}} + {{bu}(t)}}},\quad{{y(t)} = {c^{T}{x(t)}}},} & (1)\end{matrix}$Where, M,NεR^(n×n), x,b,cεR^(n) and y(t)εR^(n). Matrix M comprisescapacitance c and inductance L, matrix N comprises conductance G andresistance R, state matrix x(t) comprises node voltage and branchcurrent. And, u(t) is input signal, and y(t) is output signal. Let A=−N⁻¹M and r=N⁻¹b, formula (1) can be expressed as: $\begin{matrix}{{{A\frac{\mathbb{d}{x(t)}}{\mathbb{d}t}} = {{x(t)} - {{ru}(t)}}},\quad{{y(t)} = {c^{T}{x(t)}}},} & (2)\end{matrix}$

Model order reduction aims to reduce the order of circuit system, andreflect efficiently the reduced circuit system of original circuitsystem. The state space matrix of reduced circuit can be expressed as:$\begin{matrix}{{{\hat{A}\frac{\mathbb{d}{\hat{x}(t)}}{\mathbb{d}t}} = {{\hat{x}(t)} - {\hat{r}{u(t)}}}},\quad{{\hat{y}(t)} = {{\hat{c}}^{T}{\hat{x}(t)}}},} & (3)\end{matrix}$Where, {circumflex over (x)}(t)εR^(q), ÂεR^(q×q), {circumflex over (r)},ĉεR^(q) and q<<n.

Let X(s)=L[x(t)] and {circumflex over (X)}(s)=L[{circumflex over(x)}(t)] are pulse responses of original system and reduced system inLaplace Domain, X(s) and {circumflex over (X)}(s) can be expressed asfollows:X(s)=(I _(n) −sA)⁻¹ ,{circumflex over (X)}(s)=(I _(q) −sÂ)⁻¹ {circumflexover (r)}  (4)Where, I_(n) is unit matrix of n×n, and I_(m) is unit matrix of q×q. Thetransfer function H(s) of original system and transfer function Ĥ(s) ofreduced system can be expressed separately as:H(s)=c ^(T) X(s)=c ^(T)(I _(n) −sA)⁻¹ r  (5)AndĤ(s)=ĉ ^(T) {circumflex over (X)}(s)=ĉ ^(T)(I _(q) −sÂ)⁻¹ {circumflexover (r)}  (6)Modeling Reduction Technique

To calculate the reduced model of very-large IC interconnects,well-known non-symmetric Lanczos Algorithm (P. Feldmann and R. W.Freund, “Efficient Linear Circuit Analysis by Pade Approximation viaLanczos Process”, IEEE Trans. on CAD of ICS, Vol. 14, No. 5, 1995) isemployed to set up two projection matrixes V_(q) and W_(q), and generatereduced models by two-sided projections, with the pseudo code of thealgorithm as shown in FIG. 1. The algorithm is required to provide anorder q of reduced model. To keep the characteristic consistency ofreduced model and original system, it is required to increase the orderq. However, in order to minimize the computational complexity in systemsimulation, it is required to reduce the order q. To address theaforesaid tradeoff, the present invention attempts to improve originalnon-symmetric PVL, and judge the iteration termination conditions duringcomputation. It aims to realize a maximum accuracy nearby expandfrequency point under the lowest level of computational complexity, withthe improved flow process as shown in FIG. 2. With input parameters ofvarious passive components in original circuit in Step (102), it ispossible to establish the corresponding Modified Nodal Analysis Equationfor comparison of reduced circuit model. In Step (104), projectiontechnique of original circuit is applied to generate a reduced-ordersystem by firstly setting the order of reduced model q=1. In Step (106),non-symmetric Lanczos Algorithm is used to input matrix A and itstranspose matrix as well as two original vectors b and c′, therebyobtaining bi-orthogonal matrix V_(q)=└v₁,v₂,∀,v_(q)┘ andW_(q)=└w₁,w₂,∀,w_(q)┘, namely: W_(q)′V_(q)=I, of which IεR^(q×q).Moreover, V_(q) exists in Krylov Subspace K_(q)(A,b)=span{└b Ab A²b ∀A^(q−1)b┘}, which can be developed from the Basis of K_(q). Meanwhile,W_(q) exists in Krylov Subspace L_(q)(A′,c′)=span{└c′ (A′)c′ (A′)²c′ ∀(A′)^(q−1)c′┘}, which can be developed from the Basis of L_(q).Furthermore, every iteration process will yield new bi-orthogonalvectors v_(q) and w_(q). In addition, original system's matrix A can bereduced to a tri-diagonal matrix according to non-symmetric LanczosAlgorithm: $T_{q} = \begin{bmatrix}\alpha_{1} & \beta_{2} & \quad & \quad & \quad & \quad \\\delta_{2} & \alpha_{2} & \beta_{3} & \quad & \quad & \quad \\\quad & \delta_{3} & \alpha_{3} & \beta_{4} & \quad & \quad \\\quad & \quad & \delta_{4} & Ο & Ο & \quad \\\quad & \quad & \quad & Ο & Ο & \beta_{q} \\\quad & \quad & \quad & \quad & \delta_{q} & \alpha_{q}\end{bmatrix}$

The calculation of the aforesaid matrixes will satisfy:AV _(q) =V _(q) T _(q)+δ_(q+1) v _(q+1) e _(q)′  (7)A′W _(q) =W _(q) T _(q)′+β_(q+1) w _(q+1) e _(q)′  (8)and W_(q)′AV_(q)=T_(q), where e_(q) is q-th row vector in unit matrixIεR^(q×q).

Owing to the error of moment between reduced model and original model,formula (7) and (8) can be expressed as the following three-termrecursive equations in order to reduce efficiently the computation ofsystem analysis and minimize the error:Av _(q)=β_(q) v _(q−1)+α_(q) v _(q)+δ_(q+1) v _(q+1)  (9)A′w _(q)=δ_(q) w _(q−1)+α_(q) w _(q)+β_(q+1) v _(q+1)  (10)

Since δ_(q+1) can be treated as the component of new vector v_(q+1) inAV_(q), and β_(q+1) treated as the component of new vector w_(q+1) inA′W_(q), Step (108) selects δ_(q+1) and β_(q+1) as a referenceindicator. Similarly, Step (110) takes${\lambda_{q} = {\frac{\delta_{q + 1}}{{AV}_{q}}}},{\mu_{q} = {\frac{\beta_{q + 1}}{A^{\prime}W_{q}}}}$as an indicator for terminating the iteration process. Assuming thatλ_(q) and μ_(q) are less than tolerances for termination conditions, thereduced system will be very similar to original system. Ifabove-specified conditions are not met, the order of reduced model willbe gradually increased in Step (112). Every iteration will generate newbi-orthogonal vectors v_(q) and w_(q) as well as new δ_(q+1) andβ_(q+1). When both λ_(q) and μ_(q) meets the conditions as specified inStep (110), non-symmetric Lanczos Algorithm iteration process will bestopped, in such case q is an optimal order of reduced model. In Step(114), order q is used for reduction of system model.Addition of Perturbed System

In original interconnect circuit system, Additive Perturbation Matrixcan be added to demonstrate the reliability of this method. Suppose thatthe circuit is Modified Nodal Analysis Equation is: $\begin{matrix}{{{( {A - \Delta} )\frac{\mathbb{d}{x_{\Delta}(t)}}{\mathbb{d}t}} = {{x_{\Delta}(t)} - {{ru}(t)}}},\quad{{y_{\Delta}(t)} = {c^{T}{x_{\Delta}(t)}}}} & (11)\end{matrix}$Where, Δ represents additive perturbations of system:Δ=Δ₁+Δ₂  (12)Where, Δ₁=v_(q+1)δ_(q+1)w_(q)′, Δ₂=v_(q)β_(q+1)w_(q+1)′, and q is theorder of reduced model.

The transfer function of original circuit is reduced model can beexpressed as Ĥ(s), as shown in formula (6). Under the condition offormula (12), the transfer function H_(Δ)(S) of original system plusperturbed system will be equal to transfer function Ĥ(s) of reducedsystem. Let expend frequency point s=s₀+σ, andl′r=(β₁w₁)′δ₁v₁=β₁δ₁(w₁′v₁)=β₁δ₁, the transfer function of reduced modelcan be simplified as: $\begin{matrix}\begin{matrix}{{\hat{H}( {s_{0} + \sigma} )} = {{{\hat{l}}^{\prime}( {I_{q} - {\sigma\hat{A}}} )}^{- 1}\hat{r}}} \\{= {l^{\prime}{V_{m}( {I_{q} - {\sigma\quad T_{q}}} )}^{- 1}W_{m}^{\prime}r}} \\{= {\beta_{1}w_{1}^{\prime}{V_{q}( {I_{q} - {\sigma\quad T_{q}}} )}^{- 1}W_{q}^{\prime}v_{1}\delta_{1}}} \\{{= {( {l^{\prime}r} ){e_{1}^{\prime}( {I_{q} - {\sigma\quad T_{q}}} )}^{- 1}e_{1}}},}\end{matrix} & (13)\end{matrix}$And the transfer function of perturbed system can be simplified as:$\begin{matrix}\begin{matrix}{{H_{\Delta}( {s_{0} + \sigma} )} = {{l^{\prime}( {I_{n} - {\sigma( {A - \Delta} )}} )}^{- 1}r}} \\{= {w_{1}^{\prime}{\beta_{1}( {I_{n} - {\sigma( {A - \Delta} )}} )}^{- 1}\delta_{1}v_{1}}} \\{= {( {l^{\prime}r} ){w_{1}^{\prime}( {I_{n} - {\sigma( {A - \Delta} )}} )}^{- 1}{v_{1}.}}}\end{matrix} & (14)\end{matrix}$By using formula (7), it can be shown that various moments of reducedmodel via PVL method are equivalent to those of original model withadditive perturbed system Δ. Firstly, subtract ΔV_(q) at both sides ofthe equation, right side of the equation can be reduced as:V _(q) T _(q)+δ_(q+1) v _(q+1) e _(q) ′−ΔV _(q) =V _(q) T _(q)+δ_(q+1) v_(q+1) e _(q)′−(v _(q+1)δ_(q+1) w _(q) ′+v _(q)β_(q+1) w _(q+1)′)V _(q)=V _(q) T _(q)

Formula (7) can be rewritten as:AV _(q) −ΔV _(q) =V _(q) T _(q)+δ_(q+1) v _(q+1) e _(q) ′−ΔV _(q)(A−Δ)V_(q) =V _(q) T _(q)

If we multiply −σ and add V_(q) at both sides of the aforesaid equation,it can be rewritten as:V _(q)−(A−Δ)V _(q) =V _(q) −σV _(q) T _(q)  (15)

Thus(I _(n)−σ(A−Δ))V _(q) =V _(q)(I _(q) −σT _(q))  (16)

If we multiply (I_(n)−σ(A−Δ))⁻¹ at matrix left of the equation, andmultiply (I_(q)−σT_(q))⁻¹ at matrix right of the equation, then formula(16) can be rewritten as:V _(q)(I _(q) −σT _(q))⁻¹=(I _(n)−σ(A−Δ))⁻¹ V _(q)  (17)

Finally, multiply w₁′ at matrix left of the equation, e₁ at matrixright, and multiply constant l′r=β₁δ₁ at both sides, then:w ₁ ′V _(q)(I _(m) −σT _(q))⁻¹ e ₁ =w ₁′(I _(n)−σ(A−Δ))⁻¹ V _(q) e ₁(l′r)e ₁′(I _(q) −σT _(q))⁻¹ e ₁=(l′r)w ₁′(I _(n)−σ(A−Δ))⁻¹ v ₁  (18)

By comparing formula (19) and (13)/(14):Ĥ(s ₀+σ)=H _(Δ)(s ₀+σ)  (19)

The reduced model derived from the aforementioned equations demonstratesnon-symmetric PVL Algorithm, where various moments of transfer functionare equal to those of original system with additive perturbed system Δ.

The model reduction method of present invention for high-speedvery-large IC employs an improved non-symmetric Lanczos Algorithm, withits pseudo code shown in FIG. 3.

SIMPLE EMBODIMENT OF THE INVENTION

The present invention tests a simple embodiment in order to verify thevalidity of proposed algorithm. FIG. 4 depicts a circuit model with 12lines. The line parameters are: resistance: 1.0 Ω/cm; capacitance: 5.0pF/cm; inductance: 1.5 nH/cm; driver resistance: 3Ω, and loadcapacitance: 1.0 pF. Each line is 30 mm long and divided into 10sections. Thus, the dimension of MNA matrix is: n=238. Under a frequencyfrom 0 to 15 GHZ, it should be possible to observe the frequencyresponse of V_(out) node voltage of the embodiment, and set the expandfrequency point of reduced model s₀=0H_(Z). If non-symmetric LanczosAlgorithm is performed, record the values of β_(i+1) and δ_(i+1) intri-diagonal matrix. If you continue to enable non-symmetric LanczosAlgorithm iteration process, and set the tolerance error of terminationconditions ε=10⁻³, i.e. u_(q)<10⁻³, it is should be possible to obtainthe optimal solution of accuracy and reduce computational complexitywhen reduced model is q=14. FIG. 5 depicts the curve diagram oftermination conditions${\lambda_{q} = {\frac{\delta_{q + 1}}{{AV}_{q}}}},{\mu_{q} = {\frac{\beta_{q + 1}}{A^{\prime}W_{q}}}}$during computational process of algorithm. FIG. 6 shows the frequencyresponse diagram of the reduced model, where H(s), Ĥ(s) and H_(Δ) ⁻(s)represent respectively the transfer function of original circuit, thetransfer function of system after performing non-symmetric Lanczosorder-reduction method, and the transfer function of original circuitwith additive perturbed system. As shown in FIG. 6, the analysis ofthree perturbations involves Δ₁, Δ₂ and Δ perturbed systems. FIG. 7analyzes the error between three perturbed systems and appliednon-symmetric Lanczos Algorithm. It can thus be found that, perturbedsystem Δ will vary from the reduced-order system, and maintain ahigh-level consistency with non-symmetric Lanczos Algorithm. Asillustrated in FIG. 8, the moment values of system are observed. Withthe increase of order, it can be found that the moment values are farless than the floating accuracy of operating system (EPS, about2.22e-16). So, the error arising from inaccuracy of operational factorsmay be ignored.

In brief, the present invention has derived very-large RLC interconnect,and implemented the model reduction method by employing non-symmetricLanczos Algorithm, thereby helping to judge automatically the order ofreduced model while maintaining the accuracy and reducing computationcomplexity. At the same time, the present invention also derived that,transfer function of original circuit with additive perturbations canrepresent approximation transfer function. Of which, perturbation matrixis related to the component generated by non-symmetric LanczosAlgorithm, so the computational quantity is very small.

The above-specified, however, are only used to describe the operatingprinciple of the present invention, but not limited to its applicationrange. However, it should be appreciated that a variety of embodimentsand various modifications are embraced within the scope of the followingclaims, and should be deemed as a further development of the presentinvention.

1. A method of determining high-speed VLSI reduced-order interconnect bynon-symmetric Lanczos Algorithm, which includes the following steps: a)input a mesh circuit; b) input an expand frequency point; c) set a statespace matrix of circuit; and d) judge the order of reduced model formodel reduction as per iteration termination conditions.
 2. A high-speedVLSI reduced-order interconnect determined by non-symmetric LanczosAlgorithm defined in claim 1, wherein including iteration terminationconditions; and since δ_(q+1) can be treated as component of new vectorv_(q+1) in AV_(q), and β_(q+1) treated as component of new vectorw_(q+1) in A′W_(q), it is required to satisfy the order q of:$\lambda_{q} = {{{\frac{\delta_{q + 1}}{{AV}_{q}}} \leq {ɛ\quad{and}\quad\mu_{q}}} = {{\frac{\beta_{q + 1}}{A^{\prime}W_{q}}} \leq ɛ}}$where, ε is a sufficiently small tolerance.
 3. A high-speed VLSIreduced-order interconnect determined by non-symmetric Lanczos Algorithmdefined in claim 1, wherein the model reduction method means that thetransfer function of perturbed system is equal to transfer function H(s)of original system with additive perturbations; the transfer functionH_(Δ)(s) of modified nodal analysis can be expressed as follows:${{( {A - \Delta} )\frac{\mathbb{d}{x_{\Delta}(t)}}{\mathbb{d}t}} = {{x_{\Delta}(t)} - {{ru}(t)}}},\quad{{y_{\Delta}(t)} = {c^{T}{x_{\Delta}(t)}}}$where, Δ=v_(q+1)δ_(q+1)w_(q)′+v_(q)β_(q+1)w_(q+1)′, q is the order ofreduced model by employing non-symmetric Lanczos Algorithm, whileδ_(q+1) and β_(q+1) can be obtained from computational process ofreduced system; the transfer function H_(Δ)(s) of perturbed system isequal to transfer function Ĥ(s) of reduced system.